uti no inu hodo booru wo oikakeruno ga sukina inu ha imasen.

Here, うち means my home / our home / my family, and by extension it often means our in casual speech.

So うちの犬 means our dog or my dog.

In everyday Japanese, うち is very commonly used in this possessive way:

• うちの猫 = our cat / my cat
• うちの母 = my mother

It sounds more homey and conversational than some other ways of saying my.

The particle の links one noun to another, often showing possession or affiliation.

So:

• うち = home / our household
• うちの犬 = the dog belonging to our household

In other words, の is doing the job that English 's or of might do.

Examples:

• 私の本 = my book
• 学校の先生 = the school's teacher / a teacher at school
• 日本の文化 = Japanese culture / the culture of Japan

ほど is being used in a comparison pattern. In this sentence, it means something like as much as or to the extent of.

The important pattern is:

A ほど B はない / いない
= There is nothing / no one that is as B as A

So here:

• うちの犬ほど ... 好きな犬はいません means
• There is no dog that likes ... as much as our dog does

This is a very common way to express nothing compares to A in terms of B.

Examples:

• 彼ほど速く走れる人はいない。
  = No one can run as fast as him.
• 東京ほど人が多い町はない。
  = There is no town as crowded as Tokyo.

It is repeated because the sentence is comparing our dog with other dogs.

Structure-wise, it is basically:

• うちの犬ほど = as much as our dog
• ボールを追いかけるのが好きな犬 = a dog that likes chasing balls
• はいません = does not exist / there isn't

So the second 犬 is the noun being described by the phrase ボールを追いかけるのが好きな.

A natural way to think about it is:

There is no dog that likes chasing balls as much as our dog.

Japanese often repeats a noun where English might avoid repetition.

Because の turns the verb phrase ボールを追いかける (to chase balls) into something noun-like.

This is called nominalization.

• ボールを追いかける = chase balls
• ボールを追いかけるの = chasing balls / the act of chasing balls

Then that whole action becomes the thing that is liked:

• ボールを追いかけるのが好き
  = likes chasing balls

In English, we often use -ing for this idea:

• I like running
• She likes reading
• The dog likes chasing balls

Japanese often uses の or こと to do this.

Because 好き behaves more like a na-adjective or a descriptive word meaning liked / pleasing, rather than an action verb like English like.

In Japanese, the thing that is liked is commonly marked with が:

• 犬が好き = I like dogs
• 音楽が好き = I like music
• ボールを追いかけるのが好き = likes chasing balls

This feels strange to English speakers because English uses like + object, but Japanese uses a different structure.

So:

• English: likes X
• Japanese: X が好き

Because 好き is a na-adjective, so when it directly modifies a noun, it needs な.

So:

• 好きだ = is liked / is fond of
• 好きな犬 = a dog that likes... / a dog that is fond of...

More examples:

• 静かな部屋 = a quiet room
• 有名な人 = a famous person
• 好きな食べ物 = favorite food / food one likes

In this sentence, 好きな modifies 犬:

• ボールを追いかけるのが好きな犬
  = a dog that likes chasing balls

The は marks 犬 as the topic or contrastive topic.

So 犬はいません is not just a plain there are no dogs, but more like:

• As for dogs like that, there are none
• There is no such dog

In this sentence, は helps emphasize the comparison:

• As for a dog that likes chasing balls as much as our dog, there isn't one.

Japanese often uses は in negative comparison statements like this:

• これほどおいしい店はない。
  = There is no restaurant this delicious.
• 彼ほど親切な人はいない。
  = There is no person as kind as him.

Because いる is used for the existence of animate beings, including people and animals.

Since 犬 is an animal:

• 犬がいる = there is a dog
• 犬はいません = there is no dog

If it were an inanimate object, Japanese would use ある / ありません instead:

• 本があります = there is a book
• 本はありません = there is no book

So いません is the correct negative form here because the sentence is saying no such dog exists.

A helpful way to break it down is:

うちの犬ほど
= as much as our dog

ボールを追いかけるのが好きな犬
= a dog that likes chasing balls

はいません
= does not exist / there isn't

So the whole structure is:

[A] ほど [description] 犬 はいません
= There is no dog [with that description] as much as A

More literally:

• As for a dog that likes chasing balls to the extent of our dog, there isn't one.

That sounds unnatural in English, but grammatically it shows how the Japanese sentence is built.

Yes, こと is often possible when nominalizing a verb phrase:

• ボールを追いかけることが好き
• ボールを追いかけるのが好き

Both are grammatical.

However, の often sounds a bit more natural and immediate for concrete, familiar actions, while こと can sound slightly more formal or abstract.

In this sentence, の fits very naturally because it refers to the activity itself in a vivid, everyday way:

• chasing balls

So while both can work, の is a very natural choice here.